We present a methodology for screening predictors that, given the response, follow a one-parameter exponential family distributions. Screening predictors can be an important step in regressions when ...the number of predictors p is excessively large or larger than n the number of observations. We consider instances where a large number of predictors are suspected irrelevant for having no information about the response. The proposed methodology helps remove these irrelevant predictors while capturing those linearly or nonlinearly related to the response.
The optimization of a real-valued objective function f(U), where U is a p X d,p > d, semi-orthogonal matrix such that UTU=Id, and f is invariant under right orthogonal transformation of U, is often ...referred to as a Grassmann manifold optimization. Manifold optimization appears in a wide variety of computational problems in the applied sciences. In this article, we present GrassmannOptim, an R package for Grassmann manifold optimization. The implementation uses gradient-based algorithms and embeds a stochastic gradient method for global search. We describe the algorithms, provide some illustrative examples on the relevance of manifold optimization and finally, show some practical usages of the package.
In regression settings, a su?cient dimension reduction (SDR) method seeks the core information in a p-vector predictor that completely captures its relationship with a response. The reduced predictor ...may reside in a lower dimension d < p, improving ability to visualize data and predict future observations, and mitigating dimensionality issues when carrying out further analysis. We introduce ldr, a new R software package that implements three recently proposed likelihood-based methods for SDR: covariance reduction, likelihood acquired directions, and principal fitted components. All three methods reduce the dimensionality of the data by pro jection into lower dimensional subspaces. The package also implements a variable screening method built upon principal ?tted components which makes use of ?exible basis functions to capture the dependencies between the predictors and the response. Examples are given to demonstrate likelihood-based SDR analyses using ldr, including estimation of the dimension of reduction subspaces and selection of basis functions. The ldr package provides a framework that we hope to grow into a comprehensive library of likelihood-based SDR methodologies.
A high dimensional regression setting is considered with p predictors X = (X1,...,X p)T and a response Y. The interest is with large p, possibly much larger than n the number of observations. Three ...novel methodologies based on Principal Fitted Components models (PFC; Cook, 2007) are presented: (1) Screening by PFC (SPFC) for variable screening when p is excessively large, (2) Prediction by PFC (PPFC), and (3) Sparse PFC (SpPFC) for variable selection. SPFC uses a test statistic to detect all predictors marginally related to the outcome. We show that SPFC subsumes the Sure Independence Screening of Fan and Lv (2008). PPFC is a novel methodology for prediction in regression where p can be large or larger than n. PPFC assumes that X∣Y has a normal distribution and applies to continuous response variables regardless of their distribution. It yields accuracy in prediction better than current leading methods. We adapt the Sparse Principal Components Analysis (Zou et al., 2006) to the PFC model to develop SpPFC. SpPFC performs variable selection as good as forward linear model methods like the lasso (Tibshirani, 1996), but moreover, it encompasses cases where the distribution of Y∣X is non-normal or the predictors and the response are not linearly related.
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